The present invention relates to computed tomography (CT) imaging apparatus; and more particularly, to a method for reconstructing images from divergent beams of acquired image data.
In a current computed tomography system, an x-ray source projects a fan-shaped beam which is collimated to lie within an X-Y plane of a Cartesian coordinate system, termed the “imaging plane.” The x-ray beam passes through the object being imaged, such as a medical patient, and impinges upon an array of radiation detectors. The intensity of the transmitted radiation is dependent upon the attenuation of the x-ray beam by the object and each detector produces a separate electrical signal that is a measurement of the beam attenuation. The attenuation measurements from all the detectors are acquired separately to produce the transmission profile.
The source and detector array in a conventional CT system are rotated on a gantry within the imaging plane and around the object so that the angle at which the x-ray beam intersects the object constantly changes. A group of x-ray attenuation measurements from the detector array at a given angle is referred to as a “view” and a “scan” of the object comprises a set of views made at different angular orientations during one revolution of the x-ray source and detector. In a 2D scan, data is processed to construct an image that corresponds to a two dimensional slice taken through the object. The prevailing method for reconstructing an image from 2D data is referred to in the art as the filtered backprojection technique. This process converts the attenuation measurements from a scan into integers called “CT numbers” or “Hounsfield units”, which are used to control the brightness of a corresponding pixel on a display.
The term “generation” is used in CT to describe successively commercially available types of CT systems utilizing different modes of scanning motion and x-ray detection. More specifically, each generation is characterized by a particular geometry of scanning motion, scanning time, shape of the x-ray beam, and detector system.
As shown in FIG. 1, the first generation utilized a single pencil x-ray beam and a single scintillation crystal-photomultiplier tube detector for each tomographic slice. After a single linear motion or traversal of the x-ray tube and detector, during which time 160 separate x-ray attenuation or detector readings are typically taken, the x-ray tube and detector are rotated through 1° and another linear scan is performed to acquire another view. This is repeated typically to acquire 180 views.
A second generation of devices developed to shorten the scanning times by gathering data more quickly is shown in FIG. 2. In these units a modified fan beam in which anywhere from three to 52 individual collimated x-ray beams and an equal number of detectors are used. Individual beams resemble the single beam of a first generation scanner. However, a collection of from three to 52 of these beams contiguous to one another allows multiple adjacent cores of tissue to be examined simultaneously. The configuration of these contiguous cores of tissue resembles a fan, with the thickness of the fan material determined by the collimation of the beam and in turn determining the slice thickness. Because of the angular difference of each beam relative to the others, several different angular views through the body slice are being examined simultaneously. Superimposed on this is a linear translation or scan of the x-ray tube and detectors through the body slice. Thus, at the end of a single translational scan, during which time 160 readings may be made by each detector, the total number of readings obtained is equal to the number of detectors times 160. The increment of angular rotation between views can be significantly larger than with a first generation unit, up to as much as 36°. Thus, the number of distinct rotations of the scanning apparatus can be significantly reduced, with a coincidental reduction in scanning time. By gathering more data per translation, fewer translations are needed.
To obtain even faster scanning times it is necessary to eliminate the complex translational-rotational motion of the first two generations. As shown in FIG. 3, third generation scanners therefore use a much wider fan beam. In fact, the angle of the beam may be wide enough to encompass most or all of an entire patient section without the need for a linear translation of the x-ray tube and detectors. As in the first two generations, the detectors, now in the form of a large array, are rigidly aligned relative to the x-ray beam, and there are no translational motions at all. The tube and detector array are synchronously rotated about the patient through an angle of 180–360°. Thus, there is only one type of motion, allowing a much faster scanning time to be achieved. After one rotation, a single tomographic section is obtained.
Fourth generation scanners feature a wide fan beam similar to the third generation CT system as shown in FIG. 4. As before, the x-ray tube rotates through 360° without having to make any translational motion. However, unlike in the other scanners, the detectors are not aligned rigidly relative to the x-ray beam. In this system only the x-ray tube rotates. A large ring of detectors are fixed in an outer circle in the scanning plane. The necessity of rotating only the tube, but not the detectors, allows faster scan time.
Most of the commercially available CT systems employ image reconstruction methods based on the concepts of Radon space and the Radon transform. For the pencil beam case, the data is automatically acquired in Radon space. Therefore a Fourier transform can directly solve the image reconstruction problem by employing the well-known Fourier-slice theorem. Such an image reconstruction procedure is called filtered backprojection (FBP). The success of FBP reconstruction is due to the translational and rotational symmetry of the acquired projection data. In other words, in a parallel beam data acquisition, the projection data are invariant under a translation and/or a rotation about the object to be imaged. For the fan beam case, one can solve the image reconstruction problem in a similar fashion, however, to do this an additional “rebinning” step is required to transform the fan beam data into parallel beam data. The overwhelming acceptance of the concepts of Radon space and the Radon transform in the two dimensional case gives this approach to CT image reconstruction a paramount position in tomographic image reconstruction.
The Radon space and Radon transformation reconstruction methodology is more problematic when applied to three-dimensional image reconstruction. Three-dimensional CT, or volume CT, employs an x-ray source that projects a cone beam on a two-dimensional array of detector elements as shown in FIG. 5. Each view is thus a 2D array of x-ray attenuation measurements and a complete scan produced by acquiring multiple views as the x-ray source and detector array are revolved around the subject results in a 3D array of attenuation measurements. The reason for this difficulty is that the simple relation between the Radon transform and the x-ray projection transform for the 2D case in not valid in the 3D cone beam case. In the three-dimensional case, the Radon transform is defined as an integral over a plane, not an integral along a straight line. Consequently, we have difficulty generalizing the success of the Radon transform as applied to the 2D fan beam reconstruction to the 3D cone beam reconstruction. In other words, we have not managed to derive a shift-invariant FBP method by directly rebinning the measured cone beam data into Radon space. Numerous solutions to this problem have been proposed as exemplified in U.S. Pat. Nos. 5,270,926; 6,104,775; 5,257,183; 5,625,660; 6,097,784; 6,219,441; and 5,400,255.
It is well known that the projection-slice theorem (PST) plays an important role in the image reconstruction from two- and three-dimensional parallel-beam projections. The power of the PST lies in the fact that Fourier transform of a single view of parallel-beam projections is mapped into a single line (two-dimensional case) or a single slice (three-dimensional case) in the Fourier space via the PST. In other words, a complete Fourier space of the image object can be built up from the Fourier transforms of the sequentially measured parallel-beam projection data. Once all the Fourier information of the image object is known, an inverse Fourier transform can be performed to reconstruct the image. Along the direction of the parallel-beam projections, there is a shift-invariance of the image object in a single view of the parallel-beam projections. This is the fundamental reason for the one-to-one correspondence between the Fourier transform of parallel-beam projections and a straight line or a slice in the Fourier space. The name of the projection-slice theorem follows from this one-to-one correspondence.
In practice, divergent fan-beam and cone-beam scanning modes have the potential to allow fast data acquisition. But image reconstruction from divergent-beam projections poses a challenge. In particular, the PST is not directly applicable to the divergent-beam projections since the shift-invariance in a single view of projections is lost in the divergent-beam cases. One way to bypass this problem is to explicitly rebin the measured divergent-beam projections into parallel beam projections. This is the basic method currently used in solving the problem of fan-beam image reconstruction. After the rebinning process, one can take the advantages of the fast Fourier transforms (FFT) to efficiently reconstruct images. There are some issues on the potential loss of image spatial resolution due to the data rebinning. But there are also some advantages in generating uniform distribution of image noise due to the non-local characteristic of the Fourier transform. Alternatively, a fan-beam projection can also be relabeled in terms of Radon variables so that the two-dimensional inverse Radon transform can be used to reconstruct images. In this way, a convolution-based fan-beam image reconstruction algorithm can be readily developed. The advantage of this type of reconstruction algorithm is the explicit filtered backprojection (FBP) structure. The disadvantage of the convolution-based method is that the weight in the backprojection step depends on the individual image pixels and thus noise distribution may not be uniform. This may pose problems in the clinical interpretation of tomographic images. In practice, different CT manufactures may utilize different strategies in balancing these advantages and disadvantages.
In the cone-beam case, it is much more complicated to rebin cone-beam projections into parallel-beam projections. The huge cone-beam data set also poses a big challenge to the potential data storage during the rebinning process. The main stream of the developments in cone-beam reconstruction has been focused on the development of approximate or exact reconstruction methods. For circular-based source trajectories, methods disclosed by L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical Cone Beam Algorithm,” J. Opt. Soc. Am. A 1, 612–619(1984); G. Wang, T. H. Lin, P. Cheng, and D. M. Shinozaki, “A general cone-beam reconstruction algorithm,” IEEE Trans. Med. Imaging 12, 486–496 (1993); generate acceptable image quality up to moderate cone angles (up to 10° or so). Exact reconstruction algorithms have also been proposed and further developed for both helical source trajectory and more general source trajectories. Most recently, a mathematically exact and shift-invariant FBP reconstruction formula was proposed for the helical/spiral source trajectory A. Katsevich, “Theoretically exact filtered backprojection-type inversion algorithm for spiral CT,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 2012–2026 (2002). Starting with either the original Tuy's framework or Grangeat's framework, upon an appropriate choice of weighting function over the redundant data, shift-invariant FBP reconstruction formula has been derived for a general source trajectory. Similar to the fan-beam FBP reconstruction algorithm the characteristic of the convolution-based cone-beam reconstruction algorithm is the voxel-dependent weighting factor in the backprojection step. This will cause non-uniform distribution of the image noise. Moreover, due to the local nature of the newly developed convolution-based cone-beam image reconstruction algorithms, different image voxels are reconstructed by using cone-beam projection data acquired at different pieces of the source trajectory. Namely, different image voxels are reconstructed by using the data acquired under different physical conditions. This will potentially lead to some data inconsistency in dynamic imaging. Finally, the current convolution-based image reconstruction algorithms are only valid for some discrete pitch values in the case of helical/spiral source trajectory. This feature limits their application in a helical/spiral cone-beam CT scanner.
Therefore, it is desirable to have some alternative ways to allow a balance between computational efficiency and non-local noise distribution in the cone-beam image reconstruction. It is also important to avoid a global data rebinning procedure. Namely, a data rebinning procedure from cone-beam projection data to parallel-beam projections after the data acquisition is complete.